Measures with real spectra

Summary We solve a problem of Y. Katznelson: ifG is a locally compact abelian group and if a measure inM(G) has real spectrum, does it follow that the range of its Fourier-Stieljes transform is dense in its spectrum? We give a general construction of continuous measures with real spectrum and singular convolution powers. It is shown that the real spectrum property is strongly related to conditions of quasi-invariance under convolutors and that, in the simple case of quasi-invariance under translations, the range of the Fourier-Stieltjes transform is dense in the spectrum. However, a construction inM ₀(T) provides a negative answer to Katznelson's question.